Problem: $\dfrac{ -5v + 8w }{ -6 } = \dfrac{ -2v + 8x }{ -7 }$ Solve for $v$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -5v + 8w }{ -{6} } = \dfrac{ -2v + 8x }{ -7 }$ $-{6} \cdot \dfrac{ -5v + 8w }{ -{6} } = -{6} \cdot \dfrac{ -2v + 8x }{ -7 }$ $-5v + 8w = -{6} \cdot \dfrac { -2v + 8x }{ -7 }$ Multiply both sides by the right denominator. $-5v + 8w = -6 \cdot \dfrac{ -2v + 8x }{ -{7} }$ $-{7} \cdot \left( -5v + 8w \right) = -{7} \cdot -6 \cdot \dfrac{ -2v + 8x }{ -{7} }$ $-{7} \cdot \left( -5v + 8w \right) = -6 \cdot \left( -2v + 8x \right)$ Distribute both sides $-{7} \cdot \left( -5v + 8w \right) = -{6} \cdot \left( -2v + 8x \right)$ ${35}v - {56}w = {12}v - {48}x$ Combine $v$ terms on the left. ${35v} - 56w = {12v} - 48x$ ${23v} - 56w = -48x$ Move the $w$ term to the right. $23v - {56w} = -48x$ $23v = -48x + {56w}$ Isolate $v$ by dividing both sides by its coefficient. ${23}v = -48x + 56w$ $v = \dfrac{ -48x + 56w }{ {23} }$